Skip to main content
SpringerLink
Log in

About Nonstationary Multiresolution Analysis and Wavelets

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

The characterization of orthonormal bases of wavelets by means of convergent series involving only the mother wavelet is known, as well as the characterization of wavelets which can be constructed from a stationary multiresolution analysis or a scaling function (see for example [11] and references therein). Here we show that under some asymptotic condition, these results remain true in the nonstationary case.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Auscher, P.: Toute base d’ondelettes régulières de \({L^2(\mathbb{R})}\) est issue d’une analyse multi-résolution régulière. C. R. Acad. Sci. Paris, t. 315, série I, 1227–1230 (1992)

  2. Auscher, P.: Il n’existe pas de bases d’ondelettes régulières dans l’espace de Hardy \({H^2(\mathbb{R})}\). C. R. Acad. Sci. Paris, t. 315, série I, 769–772 (1992)

  3. Bastin F., Boigelot C.: Biorthogonal wavelets in \({H^m(\mathbb{R})}\). J. Fourier Anal. Appl. 4(6), 749–768 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bastin F., Laubin P.: Regular compactly supported wavelets in Sobolev spaces. Duke Math. J. 87(3), 481–508 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bastin F., Laubin P.: Compactly supported wavelets in Sobolev spaces of integer order. Appl. Comp. Harmon. Anal. 4(1), 51–57 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bastin, F., Laubin, P.: A walk in the theory of wavelets from \({L^2(\mathbb{R})}\) to \({H^s(\mathbb{R})}\). Rendiconti del Circolo Matematico di Palermo, Serie II, Suppl. 52, pp. 239–252 (1998)

  7. Daubechies I.: Ten Lectures on Wavelets. SIAM, Philadelphia (1992)

    Book  MATH  Google Scholar 

  8. de Boor C., DeVore R.A., Ron A.: On the construction of multivariate (pre)wavelets. Constr. Approx. 9, 123–166 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dyn N., Ron A.: Multiresolution analysis by infinitely differentiable compactly supported functions. Appl. Comp. Harmon. Anal. 2(1), 15–20 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gripenberg G.: A necessary and sufficient condition for the existence of a father wavelet. Stud. Math. 114(3), 207–226 (1995)

    MathSciNet  MATH  Google Scholar 

  11. Hernández E., Weiss G.: A First Course on Wavelets. CRC Press, Boca Raton (1996)

    Book  MATH  Google Scholar 

  12. Lemarié-Rieusset, P.G.: Existence de “fonction-père” pour les ondelettes á support compact. C. R. Acad. Sci. Paris, t. 314, série I, 17–19 (1992)

  13. Lemarié-Rieusset P.G.: Sur l’existence des analyses multi-résolutions en théorie des ondelettes. Revista Matemática Iberoamericana 8(3), 457–474 (1992)

    Article  MATH  Google Scholar 

  14. Mallat S.G.: Multiresolution approximations and wavelet orthonormal bases of \({L^2(\mathbb{R})}\). Trans. Am. Math. Soc 35, 1 (1989)

    MathSciNet  Google Scholar 

  15. Meyer Y.: Ondelettes et opérateurs, I : Ondelettes. Hermann, Paris (1990)

    MATH  Google Scholar 

  16. Novikov, I.Ya., Protasov, V.Yu., Skopina, M.A.: Wavelet Theory (Translations of Mathematical Monographs). AMS, Providence (July 2011, translated by E. Sorokina)

  17. Vonesch C., Blu T., Unser M.: Generalized Daubechies wavelet families. IEEE Trans. Signal Process 55(9), 4415–4429 (2007)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, University of Liege, Grande Traverse 12 (Bât. B37), 4000, Liege, Belgium

    F. Bastin & L. Simons

Authors
  1. F. Bastin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. Simons
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. Simons.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bastin, F., Simons, L. About Nonstationary Multiresolution Analysis and Wavelets. Results. Math. 63, 485–500 (2013). https://doi.org/10.1007/s00025-011-0212-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00025-011-0212-z

Mathematics Subject Classification (2010)

Keywords

Search

Navigation